Chapter 13

1. Section 13.1
   • Definition of a function of two variables z=f(x,y): this is really f={(x,y,z)|z=some computation involving x and y}.
   • Definition of a function of three variables w=f(x,y,z): this is really f={(x,y,z,w)|w=some computation involving x, y, and z}.
   • and we could just keep going for more variables, although rather than using such confusing variable names as w, x, y, and z we might move to something like y=f(x₁,x₂,x₃), meaning f={(x₁,x₂,x₃,y)| y=some computation involving x₁,x₂, and x₃}

   • dependent and independent variables
   • domain and range
   • usual suspects: sum, difference, product, and quotient of functions; composition of functions (this works for the composition of a function of a single variable with a function of multiple variables but not two function each of two variables)
   • graphs of functions:
     Mathematics
     FlashandMath.com contours
     Also see this Java based program.
     Wireframe
     Wireframe
   • Level curves
     FlashandMath and Excel (with a little help from tvprog.pl and, of course, Perl). Note that the tvprog.pl file will download as tvprogpl.txt. [This complication is related to restrictions in browsers related to the kinds of files they will download. Using the altered name gets around that limitation.] Once the file has been downloaded you need to change the name to tvprog.pl and Perl must be installed in order to run the program. Of course, you will want to modify the function definition at line 56 of the program. Use $x as x, $y as y, the usual operators for addition(+), subtraction(-), multiplication(*), and division(/), with a liberal use of parentheses. Remember that there is no implied multiplication so 4$x will cause an error. Also, strangely enough use log() for the natural logarithm and ** for raising to a power, as in 2**4 for 2 to the 4th power.
   • Some note should be made here that the real world may be approximated by nice functions but that does not mean that it needs to be so. Look at an example or two.
2. Section 13.2
   • Neighborhoods: In Calc I when we wanted a neighborhood around a point in the independent variable domain, say the point `x_0`, we looked at the set of values x such that `|x-x_0| < delta`. We note that the expression `|x-x_0|` gives us the distance from x to x₀. To get the same concept of a neighborhood around a point in the independent domain of a function in two variables (one that exists in 3-space) we will look at that point as `(x_0,y_0)` [really the point `(x_0,y_0,0)` in 3-space] and we want to look at the neighbrhood of that point as the set of all points `(x,y)` [really, `(x,y,0)`] such that the distance between such a point and the point `(x_0,y_0)` [really `(x_0,y_0,0)`] is less than some value `delta`. But that distance is `sqrt((x-x_0)^2+(y-y_0)^2)` [really `sqrt((x-x_0)^2+(y-y_0)^2 +(0-0)^2)`]. This gives us a "disk" around the "point". Since the expression uses the < we are talking about an open disk around that point.
   • Interior point of a region: we can find some `delta` such that the δ-neighborhood around the point is entirely within the region.
   • Boundary point: every δ-neighborhood of the point contains both points in the region and out of the region.
   • If every point of the region R is an interior point then R is called open.
   • If the region contains all of its boundary points then it is a closed region.
   • Limit: `lim_((x,y)\to(x_0,y_0)) f(x,y) = L` if for each `epsilon > 0` there corresponds a `delta > 0` such that `|f(x,y)-L| < epsilon` whenever `0 < sqrt((x-x_0)^2 +(y-y_0)^2) < delta`.
   • Limits do not have to exist. Often we see that approaching to a point frm different directions may yield different values.
   • We could look at his by actually trying dfferent values. Having a program to help us do this would be a big aide. Approach is a program that will allow us to do just that. We could use it to look at the function `( (x^2-y^2)/(x^2+y^2))^2` coming at (0,0) along the x-axis and along the y-axis.
   • Continuity: as usual, the limit must exist and the limit must be equal to the value of the function at the point.
   • Usual ruls for the continuity of the sum, difference, product, and quotient of contnuous functions.
   • Usual rule for the continuity of a composition of a function of one variable and a function of many variables.
   • Going beyond a function of two variables...
     extend the idea of neighborhood
     prepare for a function of three variables but where one variable is a constant as a level surface in three-space.
3. Section 13.3
   • Definition of Partial Derivatives f_x as lim as Δx→0, and f_y as lim as Δy→0
     `f_x(x,y)=lim_(Deltax\to0)(f(x+Deltax,y)-f(x,y))/(Deltax)` and
     `f_y(x,y)=lim_(Deltay\to0)(f(x,y+Deltay)-f(x,y))/(Deltay)`
   • Return to our
     z=(some expression of x and y) vs.
     f(x,y)=(some expression of x and y) vs.
     f={(x,y,z)|z=(some expression of x and y)}
     to note that for `z = f(x,y)` we have
     `del/(delx)f(x,y) = f_x(x,y)=z_x =(delz)/(delx)` and
     `del/(dely)f(x,y) = f_y(x,y)=z_y =(delz)/(dely)`
     Evaluating the first partial at a point `(a,b)`
     `(delz)/(delx)|_((a,b)) = f_x(a,b)` and
     `(delz)/(dely)|_((a,b)) = f_y(a,b)`
   • Partials with more than two independent variables.
   • Second order partial derivatives
   • More nomenclature:
     `del/(delx)((delf)/(delx))=(del^2f)/(delx^2) =f_(x x)`
     `del/(dely)((delf)/(dely))=(del^2f)/(dely^2) =f_(y y)`
     `del/(dely)((delf)/(delx))=(del^2f)/(delydelx) =f_(x y)` note subtile change, and
     `del/(delx)((delf)/(dely))=(del^2f)/(delxdely) =f_(y x)` note subtile change
   • The very fortunate equality of mixed partials
     f_xy(x,y) = f_yx(x,y).
4. Section 13.4
   • review differentials given the strange nomenclature
     

     Go from:	Δy  Δx	≈ f ′(c)	to	Δy ≈ f ′(c) Δx   to   dy = f ′(c) dx
     where the separate dy and dx are not identical to the single symbol dy/dx.

   • used in approximations and in error analysis.
   • definition of total differential; `dx=Deltax` and `dy=Deltay` and the total differential of the dependent variable `z` is `dz= (delz)/(delx)dx + (delz)/(dely)dy = f_x(x,y)dx +f_y(x,y)dy`.
   • Definition of Differentiability
     given z=f(x,y), then if at a point (x₀,y₀) we can write
     Δz = f_x(x₀,y₀)Δx + f_y(x₀,y₀)Δy + ε₁Δx + ε₁Δy
     where both ε₁→0 and ε₂→0 as (Δx,Δy)→(0,0).
   • If f_x and f_y are continuous in an open region then f is differentiable on that region.
   • Using the total differential to approximate a change in the value of a function from point `(x,y)` to point `(x+Delta x, y+Delta y)`
     we know `Deltaz =((delz)/(delx))Deltax+((delz)/(dely))Deltay +epsilon_1Deltax +epsilon_2Deltay`
     then for small changes in x and y we use `Deltaz ~~dz` so
     `dz~~((delz)/(delx))Deltax+((delz)/(dely))Deltay`
   • Page showing an example of a function, its associated images, and even a model.
   • If a function of x and y is differentiable at `(x_0,y_0)` then it is continuous at that point.
5. Section 13.5
   • Chain rule: Given w=f(x,y), x=g(t), y=h(t)

     dw  dt

     =

     ∂w  ∂x

     dx  dt

     +

     ∂w  ∂y

     dy  dt

   • Extending that to w = f(x₁, x₂, x₃, ...,x_n)
   • Then extend to w=f(x,y), x=g(s,t), y=h(s,t),

     dw  ds

     =

     ∂w  ∂x

     ∂x  ∂s

     +

     ∂w  ∂y

     ∂y  ∂s

     and

     dw  dt

     =

     ∂w  ∂x

     ∂x  ∂t

     +

     ∂w  ∂y

     ∂y  ∂t

   • We can use these "chain rules" or we could actually just substitute the given functions and solve the problems normally.
   • Implicit Partial Differentiation
     assuming that we have x and y related as F(x,y)=0, and we assume that there is f such that y=f(x) and f is differentiable, (we want this so that we know that it makes sense to look for dy/dx) then take w=F(x,y)=F(x,f(y)), so
     `(dw)/(dx) = F_x(x,y) dx/dx + F_y(x,y) dy/dx`
     and since we had `w=F(x,y) = 0 ` we have `(dw)/(dx)=0`
     and we know that`dx/dx=1` and, then we have,
     `dy/dx = - (F_x(x,y))/(F_y(x,y))`
     Here is a link to an example.
6. Section 13.6
   • Directional Derivative of f, a function of two variables, in the direction specified by a line, y=mx+b, in the xy-plane. (This would give rise to an ugly limit assuming we had to try this.)
   • Directional Derivative of f, a function of two variables, in the direction specified by a line, x=g(t) and y=h(t), in the xy-plane. This allows us to use the limit definition of the derivative as t approaches 0.
   • Directional Derivative of f, a function of two variables, in the direction of u, a unit vector given as cos(θ)i + sin(θ)j:
     D_uf(x,y) = f_x(x,y) cos(θ) + f_y(x,y) sin(θ).
   • Gradient of f
     f(x,y)= f_x(x,y) i + f_y(x,y) j
   • D_uf(x,y) = f(x,y) • u
     D_uf(x,y) =||f(x,y) || ||u|| cos(φ)
     D_uf(x,y) =||f(x,y) || cos(φ)
   • Properties of gradient
     1. If f(x,y)=0, then D_uf(x,y) =0 for all u
     2. The direction of maximum increase of f is given by f(x,y). The maximum value of D_uf(x,y) is ||f(x,y)||.
     3. The direction of minimum increase of f is given by –f(x,y). The minimum value of D_uf(x,y) is – ||f(x,y)||.
   • If `gradf(x_0,y_0) != bb0` then `grad f(x_0,y_0)` is normal to the level curve at `(x_0,y_0)`.
   • Extend to a function of three variables
     for f(x,y,z) and u = ai + bj + ck as a unit vector,
     D_uf(x,y,z) = a*f_x(x,y,z) + b*f_y(x,y,z) + c*f_z(x,y,z)
     and f(x,y,z) = f_x(x,y,z) i + f_y(x,y,z) j + f_z(x,y,z) k
     so D_uf(x,y,z) = f(x,y,z) • u
     and the same properties follow.
7. Section 13.7
   • Start with a surface defined as z=f(x,y)
   • Morph that to F(x,y,z) = f(x,y) - z
   • Then look at F(x,y,z) = 0, as a level surface of F, remembering that a level surface is the set of points that give F the same, designated, value, in this case 0.
   • Small aside, What would we do with F(x,y,z) = 7?
   • S a surface given by F(x,y,z)=0, P=(x₀,y₀,z₀) a point on S. C be any curve on S through P where we have a vector function r(t)=x(t)i+y(t)j+z(t)k to define that curve, then since all points on the curve are on S, F at each point of the curve is 0, or F(x(t),y(t),z(t))=0 for all values of t.
     F′(t)=0
     F′(t)=F_x(x,y,z)x′(t) + F_y(x,y,z)y′(t) + F_z(x,y,z)z′(t)
     so at (x₀,y₀,z₀) we have 0=F(x₀,y₀,z₀) • r′(t₀)
   • If we have a surface S determined by F(x,y,z)=0 and a point P=(x₀,y₀,z₀) and we have that F(x₀,y₀,z₀)≠0, then
     The plane through P that is normal to F(x₀,y₀,z₀) is called the tangent plane to S at P, and
     The line through P having the direction F(x₀,y₀,z₀) is called the normal line to S at P.
   • Equation of Tangent Plane at (x₀,y₀,z₀) is
     F_x(x₀,y₀,z₀)(x-x₀) + F_y(x₀,y₀,z₀)(y-y₀) + F_z(x₀,y₀,z₀)(z-z₀) = 0.
   • Symmetric equation of normal line at `(x_0,y_0,z_0)` is
     `(x - x_0)/(F_x(x_0,y_0,z_0)) = (y - y_0)/(F_y(x_0,y_0,z_0)) = (z - z_0)/(F_z(x_0,y_0,z_0)) `
   • Angle of inclination :
     `cos(theta)= |gradF(x_0,y_0,z_0)*bbk|/||gradF(x_0,y_0,z_0)||`
8. Section 13.8
   • Extreme value(s) on closed bounded region
     at least one max
     at least one min.
   • Relative Extrema (all values in an open disk around the point, and note ≥ and ≤
     relative min at (x₀,y₀) if f(x,y) ≥ f(x₀,y₀)
     relative max at (x₀,y₀) if f(x,y) ≤ f(x₀,y₀)
   • Critical point
     f_x(x₀,y₀) = 0 and f_y(x₀,y₀) = 0
     or
     either does not exist at that point.
   • Second Partials test
     at a critical point form the value
     d = f_xx(a,b)f_yy(a,b) – [f_xy(a,b)]²
     1. If d>0 and f_xx(a,b)>0, then relative minimum at (a,b, f(a,b) )
     2. If d>0 and f_xx(a,b)<0, then relative maximum at (a,b, f(a,b) )
     3. If d<0 then saddle point at (a,b, f(a,b))
     4. If d=0, test is inconclusive.
9. Section 13.9
   • Linear Regression.
   • For n ordered pairs `(x_i,y_i)` the linear least squares regression in the form `y=ax + b` is given by setting the coefficients to For `f(x)=ax+b`, we define `S(a,b) = Sigma_(i=1)^n((F(x_i)-y_i)^2) = Sigma_(i=1)^n((ax_i+b-y_i)^2)`

   • `S_a(a,b) = Sigma_(i=1)^n(2x_i(ax_i+b-y_i))`

     `S_a(a,b) = 2aSigma_(n=1)^n(x_i^2) + 2bSigma_(i=1)^n(x_i) -2Sigma_(i=1)^n(x_iy_i)`

   • `S_b(a,b) = Sigma_(i=1)^n(2(ax_i+b - y_i)) `

     `S_b(a,b) = 2aSigma_(i=1)^n(x_i) +2nb -2Sigma_(i=1)^n(y_i)`.

   • Given `n` pairs of points `(x_i,y_i)` we can find the least squares regression line given in the form `y = ax + b` by computing
     `a=( n sum_(i=1)^n(x_iy_i) -sum_(i=1)^n(x_i)sum_(i=1)^n(y_i))/(nsum_(i=1)^n(x_i^2) -(sum_(i=1)^n(x_i))^2) `
     and
     `b= (1/n)(sum_(i=1)^n(y_i) - asum_(i=1)^n(x_i))`
   • We could express this as finding the `a` and `b` in `y=ax+b` by solving the system of linear equations given as
     `nb + S_xa = S_y` and
     ` S_xb + S_(x^2)a = S_(xy)`
     where we understand that `n` is the number of pairs of values, `S_x` is the sum of the `x` values, `S_(x^2)` is the sum of the squared `x` values, and `S_(xy)` is the sum of the product of the values in each pair.
   • Linear Regression: see Regression Example and Doing a Linear Regression on a TI-83+.
10. Section 13.10: Legrange multipliers
    • Lagrange's Theorem: Let `f` and `g` have continuous first partial derivatives such that `f` has an extremum at a point `(x_0,y_0)` on the smooth constraint curve `g(x,y) = c`. If `gradg(x_0,y_0) != bb0`, then there is a real number `lambda` such that
      `gradf(x_0,y_0) = lambda gradg(x_0,y_0)`.
    • With three variables we end up with
      `gradf(x_0,y_0,z_0) = lambda gradg(x_0,y_0,z_0)`.
    • With two constraints `g` and `h` we end up with two multipliers, `lambda` and `mu` and we have
      `gradf = lambdagradg + mugradh`.
    • See the classic truncated room problem and the more elaborate classic truncated room problem.
    • Here is a 10 minute, 43 second pencast of solving a specific example of this kind of problem.